Set origin to vertex in edit mode?
In Edit mode, make sure you enable vertex selection.  

  

Click the vertex to which you want to set the origin.  

While still in Edit Mode, Mesh > Snap > Cursor to Selected 
If all you want to do is change the pivot point, set it to 3D cursor.  

  

If you actually want to change the origin.

Tab to Object mode, then Object > Transform > Origin to 3D Cursor.

Since Blender 2.8 it's Object > Set Origin > Origin to 3D Cursor.